A002025 Smaller of an amicable pair: (a,b) such that sigma(a) = sigma(b) = a+b, a < b.

220, 1184, 2620, 5020, 6232, 10744, 12285, 17296, 63020, 66928, 67095, 69615, 79750, 100485, 122265, 122368, 141664, 142310, 171856, 176272, 185368, 196724, 280540, 308620, 319550, 356408, 437456, 469028, 503056, 522405, 600392, 609928

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

Sometimes called friendly numbers, but this usage is deprecated.

All terms are abundant (A005101). - Michel Marcus, Mar 10 2013

See A125490-A125492 and A137231 for amicable triples, A036471-A036474 and A116148 for amicable quadruples, and A233553 for amicable quintuples. - M. F. Hasler, Dec 14 2013

This sequence is strictly increasing (and A002046, which contains the larger (deficient) number in each pair, is sorted by this sequence). - Jeppe Stig Nielsen, Jan 27 2015

For the related amicable pairs see A259180. - Omar E. Pol, Jul 15 2015

Pomerance (1981) shows that there are at most x*exp(-log(x)^(1/3)) terms of this sequence up to x. In particular, as originally demonstrated by Erdős, this sequence has density 0. - Charles R Greathouse IV, Aug 17 2017

References

• Mariano Garcia, Jan Munch Pedersen and Herman te Riele, Amicable pairs - a survey, pp. 179-196 in: Alf van der Poorten and Andres Stein (eds.), High Primes and Misdemeanours: Lectures in Honour of the 60th Birthday of Hugh Cowie Williams, Fields Institute Communications, AMS, Providence RI, 2004.
• Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, pages 48-49.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe and Sergei Chernykh, Table of n, a(n) for n = 1..415523 [All terms up to 10^17. Terms 39375 through 415523 were computed by Sergei Chernykh]
• J. Alanen, O. Ore and J. Stemple, Systematic computations on amicable numbers, Math. Comp., 21 (1967), 242-245.
• J. Bell, A translation of Leonhard Euler's..., arXiv:math/0409196 [math.HO], 2004-2009.
• W. Borho and H. Hoffmann, Breeding Amicable Numbers in Abundance, Math. Comp., 46 (1986), 281-293.
• S. Chernykh, Amicable pairs list.
• Paul Erdős, On amicable numbers, Publ. Math. Debrecen 4 (1955), pp. 108-111.
• E. B. Escott, Amicable numbers, Scripta Mathematica, 12 (1946), 61-72. [Annotated scanned copy]
• L. Euler, De numeris amicabilibus, Opuscula varii argumetii, pages 23-107, 1750. Reprinted in Opera mathematica: Series prima. Volumen II, Leonhardi Euleri commentationes arithmeticae. Sub ausp. soc. scient. nat. Helv., Teubner, Leipzig, Series I, Vol. 1915, pp. 86-162.
• M. Garcia, A Million New Amicable Pairs, J. Integer Seqs., Vol. 4 (2001), #01.2.6.
• Mariano García, Jan Munch Pedersen, and Herman J. J. te Riele, Amicable pairs, a survey, Report MAS-R0307, 2003, Centrum Wiskunde en Informatica.
• Mariano García, Jan Munch Pedersen, and Herman J. J. te Riele, Amicable pairs, a survey, Fields Institute Comm. (2004) Vol. 41.
• S. S. Gupta, Amicable Numbers.
• E. J. Lee, Amicable Numbers and the Bilinear Diophantine Equation, Math. Comp., 22 (1968), 181-187.
• Hisanori Mishima, First 236 amicable pairs.
• D. Moews, Perfect, amicable and sociable numbers.
• Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
• J. O. M. Pedersen, Known Amicable Pairs. [Broken link]
• J. O. M. Pedersen, Tables of Aliquot Cycles. [Broken link]
• J. O. M. Pedersen, Tables of Aliquot Cycles. [Via Internet Archive Wayback-Machine]
• J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]
• Carl Pomerance, On the distribution of amicable numbers, J. reine angew. Math. 293/294 (1977), pp. 217-222.
• Carl Pomerance, On the distribution of amicable numbers, II, J. reine angew. Math. 325 (1981), pp. 183-188.
• H. J. J. te Riele, Four large amicable pairs, Math. Comp., 28 (1974), 309-312.
• H. J. J. te Riele, Computation of all the amicable pairs below 10^10, Math. Comp., 47 (1986), 361-368 and Supplement pp. S9-S40.
• H. J. J. te Riele et al., Table of Amicable Pairs between 10^10 and 10^52, Note NM-N8603, Department of Numerical Mathematics, Centre for Mathematics and Computer Science, Amsterdam, 1986. (Warning: file size is 65MB.)
• T. Trotter, Jr., Amicable Numbers, archived from the original.
• Eric Weisstein's World of Mathematics, Amicable Pair.

Crossrefs

Cf. A000203, A002046, A063990, A066873, A180164, A259180.

Programs

Mathematica

Reap[For[n = 1, n <= 10^6, n++, If[(s = DivisorSigma[1, n]) > 2n && DivisorSigma[1, s - n] == s, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 09 2015, after M. F. Hasler *)

PARI

aliquot(n)=sigma(n)-n
isA002025(n)={if (n>1, local(a);a=aliquot(n);a>n && aliquot(a)==n)} \\ Michael B. Porter, Apr 11 2010

PARI

for(n=1,1e6,(s=sigma(n))>2*n && sigma(s-n)==s && print1(n",")) \\ M. F. Hasler, Dec 14 2013

PARI

forfactored(n=1,10^6, t=sigma(n[2])-n[1]; if(t>n[1] && sigma(t)==n[1]+t, print1(n[1]", "))) \\ Charles R Greathouse IV, Aug 17 2017

Formula

a(n) = A259180(2n-1) = A180164(n) - A259180(2n) = A180164(n) - A002046(n). - Omar E. Pol, Jul 15 2015

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Oct 24 2000